Number Systems

5√2

Step 1: Simplify each square root: √18 = √(9×2) = 3√2, √8 = √(4×2) = 2√2. Step 2: Substitute these values: 3√2 + 2√2 - √2. Step 3: Combine like terms: (3 + 2 - 1)√2 = 4√2. Step 4: Wait, let me recalculate: 3√2 + 2√2 - √2 = (3 + 2 - 1)√2 = 5√2. Therefore, the answer is 5√2.

8

Step 1: Express 16^(3/4) using the property a^(m/n) = (a^(1/n))^m. Step 2: 16^(3/4) = (16^(1/4))^3. Step 3: Find 16^(1/4): Since 2^4 = 16, we have 16^(1/4) = 2. Step 4: Therefore, 16^(3/4) = (2)^3 = 8.

(√5 + √3)/2

Step 1: Multiply numerator and denominator by (√5 + √3). Step 2: Numerator becomes 1 × (√5 + √3) = √5 + √3. Step 3: Denominator becomes (√5 - √3)(√5 + √3) = (√5)² - (√3)² = 5 - 3 = 2. Step 4: Therefore, 1/(√5 - √3) = (√5 + √3)/2.

√15

Step 1: Check each option for rationality. Step 2: 0.25 = 1/4 (rational), √16 = 4 (rational), 7/3 is already in p/q form (rational). Step 3: √15 cannot be simplified to a perfect square and cannot be expressed as p/q. Step 4: Therefore, √15 is irrational.